> [!NOTE] Definition 1 (Linear Continuity) > A function $f:\mathbb{R}^n\to \mathbb{R}^{k}$ is linearly continuous at $x_{0}$ if the $f:L\to \mathbb{R}^k$ where $L$ is a [[Line in Real n-Space|line]] passing through $x_{0}$ is continuous. > > That is, if the line $L$ is parametrised by $t\mapsto x_{0}+vt$, the function $\phi: t \mapsto f(x_{0}+ vt)$ is continuous at $t = 0$ ($\lim_{ t \to 0 }f(x_{0}+tv)=f(x_{0})$) A function may still not be continuous at a point even though it is continuous along any line passing through that point. For example, take $f(x,y) = \begin{cases} 1, & 0<y< x^{2} \\ 0, & \text{otherwise.} \end{cases}$ Since $\lim_{ x \to 0 }f\left( x, \frac{1}{2 }x^{2} \right) = 1$ while $\lim_{ x \to 0 }f(x, mx)=0$ for all $m$ (draw a picture!), this shows that $f$ is discontinuous at $(0,0)$. # Properties(s) # Application(s) **More examples**: # Bibliography